Question

. What is the value of 25+106√−−−−−−−−√+25−106√−−−−−−−−√25+106+25−106 ?

Answer 1

the of following question is 
(x+y)2=x2+2xy+y2(x+y)2=x2+2xy+y2 (while (x−y)2=x2−2xy+y2(x−y)2=x2−2xy+y2). 

So we get: (25+106√−−−−−−−−√+25−106√−−−−−−−−√)2=(25+106√−−−−−−−−√)2+2(25+106√−−−−−−−−√)(25−106√−−−−−−−−√)+(25−106√−−−−−−−−√)2=(25+106+25−106)2=(25+106)2+2(25+106)(25−106)+(25−106)2=
=(25+106√)+2(25+106√−−−−−−−−√)(25−106√−−−−−−−−√)+(25−106√)=(25+106)+2(25+106)(25−106)+(25−106). 

Note that sum of the first and the third terms simplifies to (25+106√)+(25−106√)=50(25+106)+(25−106)=50, so we have 50+2(25+106√−−−−−−−−√)(25−106√−−−−−−−−√)50+2(25+106)(25−106) --> 50+2(25+106√−−−−−−−−√)(25−106√−−−−−−−−√)=50+2(25+106√)(25−106√)−−−−−−−−−−−−−−−−−−−√50+2(25+106)(25−106)=50+2(25+106)(25−106). 

Also must know for the GMAT: (x+y)(x−y)=x2−y2(x+y)(x−y)=x2−y2, thus 50+2(25+106√)(25−106√)−−−−−−−−−−−−−−−−−−−√=50+2252−(106√)2)−−−−−−−−−−−−√=50+2625−600−−−−−−−−√=50+225−−√=6050+2(25+106)(25−106)=50+2252−(106)2)=50+2625−600=50+225=60.

Recall that we should un-square this value to get the right the answer: 60−−√=215−−√60=215.

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Answer 2

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